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Question 17 - CBSE Class 12 Sample Paper for 2024 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by

The set of all points where the function f(x)=x+|x| is differentiable, is

(a) (0,∞)                                    (b) (-∞,0)

(c) (-∞,0)∪(0,∞)                        (d) (-∞,∞)

This question is similar to Question-15 NCERT-Exemplar-MCQs

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Transcript

๐‘“(๐‘ฅ) = ๐‘ฅ+|๐‘ฅ| = {โ–ˆ(๐‘ฅ+๐‘ฅ, ๐‘ฅโ‰ฅ0@๐‘ฅโˆ’๐‘ฅ, ๐‘ฅ <0)โ”ค = {โ–ˆ(2๐‘ฅ, ๐‘ฅโ‰ฅ0@0, ๐‘ฅ<0)โ”ค Now, f(x) is a differentiable at x = 0 if LHD = RHD (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™) โˆ’ ๐’‡(๐’™ โˆ’ ๐’‰))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0) โˆ’ ๐‘“(0 โˆ’ โ„Ž))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐ŸŽ) โˆ’ ๐’‡(โˆ’๐’‰))/๐’‰ =(๐‘™๐‘–๐‘š)โ”ฌ(โ„Žโ†’0) (2(0) โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 0/โ„Ž = ๐ŸŽ (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’™ + ๐’‰) โˆ’ ๐’‡(๐’™ ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (๐‘“(0 + โ„Ž) โˆ’ ๐‘“(0))/โ„Ž = (๐’๐’Š๐’Ž)โ”ฌ(๐กโ†’๐ŸŽ) (๐’‡(๐’‰) โˆ’ ๐’‡(๐ŸŽ))/๐’‰ = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) (2โ„Ž โˆ’ 0)/โ„Ž = (๐‘™๐‘–๐‘š)โ”ฌ(hโ†’0) 2โ„Ž/โ„Ž = ๐Ÿ Hence, we can say that f(x) is differentiable on R โˆ’ {๐ŸŽ} or (โˆ’โˆž,๐ŸŽ)โˆช(๐ŸŽ,โˆž) So, the correct answer is (c)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.